Question

The solution set of the equation $2{\sin }^{2}x+\sqrt{3}\cos x+1=0$ is $\{2n\pi \pm \frac{a\pi }{b},n\in Z;\frac{a}{b}\in [0,1\left]\right\}$ where $a,b$ are coprime, then the value of $a+b$ is

Answer 1

$2{\sin }^{2}x+\sqrt{3}\cos x+1=02(1-{\cos }^{2}x)+\sqrt{3}\cos x+1=0\Rightarrow 2{\cos }^{2}x-\sqrt{3}\cos x-3=0\Rightarrow (2\cos x+\sqrt{3})(\cos x-\sqrt{3})=0$
$\Rightarrow \cos x=-\frac{\sqrt{3}}{2}$ or $\cos x=\sqrt{3}$(not possible)
$\Rightarrow \cos x=\cos \frac{5\pi }{6}\Rightarrow x\in \{2n\pi \pm \frac{5\pi }{6},n\in Z\}\therefore a=5,b=6\Rightarrow a+b=11$